Problem: All of the 3rd grade teachers and students from Gardner Bullis went on a field trip to an art museum. Tickets were $$7.50$ each for teachers and $$2.50$ each for students, and the group paid $$37.50$ in total. The next month, the same group visited a natural history museum where the tickets cost $$15.00$ each for teachers and $$10.00$ each for students, and the group paid $$120.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7.5x+2.5y = 37.5}$ ${15x+10y = 120}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-15x-5y = -75}$ ${15x+10y = 120}$ Add the top and bottom equations together. $ 5y = 45 $ $ y = \dfrac{45}{5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {7.5x+2.5y = 37.5}$ to find $x$ ${7.5x + 2.5}{(9)}{= 37.5}$ $7.5x+22.5 = 37.5$ $7.5x = 15$ $x = \dfrac{15}{7.5}$ ${x = 2}$ You can also plug ${y = 9}$ into $ {15x+10y = 120}$ and get the same answer for $x$ ${15x + 10}{(9)}{= 120}$ ${x = 2}$ There were $2$ teachers and $9$ students on the field trips.